Cube Roots of Unity and the Equation

The question asks for the solutions (\(z\)) to the equation \((z-100)^3 + 1000 = 0\). We are given that \(\omega \ne 1\) is a cube root of unity.

Key properties of the cube roots of unity (\(1, \omega, \omega^2\)) are:

• The cube roots satisfy the equation \(x^3 = 1\).
• They follow the rules: \(\omega^3 = 1\) and \(1 + \omega + \omega^2 = 0\).

The value \(\omega\) represents one of the complex cube roots of unity, often \(e^{i 2\pi/3}\) or \(e^{i 4\pi/3}\).

Solving the Cubic Equation

First, we rearrange the given equation to isolate the cubic term:

\((z-100)^3 + 1000 = 0\)

Subtract 1000 from both sides:

\((z-100)^3 = -1000\)

To make the equation easier to work with, let's use a substitution. Let \(X = z - 100\). The equation now becomes:

\(X^3 = -1000\)

Finding the Cube Roots of -1000

We need to find the three values of \(X\) that satisfy \(X^3 = -1000\). We know that \((-10)^3 = -1000\), so one solution for \(X\) is \(X = -10\).

The three cube roots of a number \(C\) can be expressed using the principal cube root and the cube roots of unity. If \(\sqrt[3]{C}\) is the principal cube root, the three roots are \(\sqrt[3]{C}\), \(\sqrt[3]{C}\omega\), and \(\sqrt[3]{C}\omega^2\).

In this case, \(C = -1000\), and the principal cube root is \(-10\). Therefore, the three possible values for \(X\) are:

• \(X_1 = -10\)
• \(X_2 = -10 \times \omega = -10\omega\)
• \(X_3 = -10 \times \omega^2 = -10\omega^2\)

Deriving the Solutions for z

Now, we need to find the values of \(z\) by substituting back \(X = z - 100\) for each of the three possible values of \(X\) found above.

1. Case 1: \(X_1 = -10\)

   \(z - 100 = -10\)

   To find \(z\), add 100 to both sides:

   \(z = 100 - 10\)

   \(z = 90\)

2. Case 2: \(X_2 = -10\omega\)

   \(z - 100 = -10\omega\)

   Add 100 to both sides:

   \(z = 100 - 10\omega\)

   We can factor out 10 from the terms on the right side:

   \(z = 10(10 - \omega)\)

3. Case 3: \(X_3 = -10\omega^2\)

   \(z - 100 = -10\omega^2\)

   Add 100 to both sides:

   \(z = 100 - 10\omega^2\)

   Factor out 10:

   \(z = 10(10 - \omega^2)\)

Combining the results from the three cases, the solutions (\(z\)) for the equation \((z-100)^3 + 1000 = 0\) are \(90\), \(10(10 - \omega)\), and \(10(10 - \omega^2)\).